Optimal. Leaf size=83 \[ -\frac {3 i (1+i \tan (c+d x))^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-i \tan (c+d x))\right )}{2^{2/3} d \sqrt {a+i a \tan (c+d x)} \sqrt [3]{e \sec (c+d x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3505, 3523, 70, 69} \[ -\frac {3 i (1+i \tan (c+d x))^{2/3} \text {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{3},\frac {5}{6},\frac {1}{2} (1-i \tan (c+d x))\right )}{2^{2/3} d \sqrt {a+i a \tan (c+d x)} \sqrt [3]{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\left (\sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}\right ) \int \frac {1}{\sqrt [6]{a-i a \tan (c+d x)} (a+i a \tan (c+d x))^{2/3}} \, dx}{\sqrt [3]{e \sec (c+d x)}}\\ &=\frac {\left (a^2 \sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-i a x)^{7/6} (a+i a x)^{5/3}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [3]{e \sec (c+d x)}}\\ &=\frac {\left (a \sqrt [6]{a-i a \tan (c+d x)} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{5/3} (a-i a x)^{7/6}} \, dx,x,\tan (c+d x)\right )}{2\ 2^{2/3} d \sqrt [3]{e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=-\frac {3 i \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{2/3}}{2^{2/3} d \sqrt [3]{e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 95, normalized size = 1.14 \[ \frac {12 i-\frac {30 i e^{2 i (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {4}{3};-e^{2 i (c+d x)}\right )}{\left (1+e^{2 i (c+d x)}\right )^{5/6}}}{16 d \sqrt {a+i a \tan (c+d x)} \sqrt [3]{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ \frac {2^{\frac {1}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (-12 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 27 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 8 \, {\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (2 i \, d x + 2 i \, c\right )}\right )} {\rm integral}\left (\frac {2^{\frac {1}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (-45 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 60 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}}{16 \, {\left (a d e e^{\left (6 i \, d x + 6 i \, c\right )} - 2 \, a d e e^{\left (4 i \, d x + 4 i \, c\right )} + a d e e^{\left (2 i \, d x + 2 i \, c\right )}\right )}}, x\right )}{8 \, {\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x +c \right )\right )^{\frac {1}{3}} \sqrt {a +i a \tan \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{1/3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{e \sec {\left (c + d x \right )}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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